caa a22 4500 475839374 CHVBK 20180406123857.0 cr unu---uuuuu 170329e20000901xx s 000 0 eng 10.1023/A:1010738827855 doi (NATIONALLICENCE)springer-10.1023/A:1010738827855 Bell Numbers, Log-Concavity, and Log-Convexity [Elektronische Daten] [Nobuhiro Asai, Izumi Kubo, Hui-Hsiung Kuo] Let {b k (n)} n=0 ∞ be the Bell numbers of order k. It is proved that the sequence {b k (n)/n!} n=0 ∞ is log-concave and the sequence {b k (n)} n=0 ∞ is log-convex, or equivalently, the following inequalities hold for all n⩾0, $$1 \leqslant \frac{{b_k (n + 2)b_k (n)}}{{b_k (n + 1)^2 }} \leqslant \frac{{n + 2}}{{n + 1}}$$ . Let {α(n)} n=0 ∞ be a sequence of positive numbers with α(0)=1. We show that if {α(n)} n=0 ∞ is log-convex, then α(n)α(m)⩽α(n+m), ∀n,m⩾0. On the other hand, if {α(n)/n!} n=0 ∞ is log-concave, then $$\alpha (n + m) \leqslant \left( {\begin{array}{*{20}c} {n + m} \\ n \\ \end{array} } \right)\alpha (n)\alpha (m),{\text{ }}\forall n,m \geqslant 0$$ . In particular, we have the following inequalities for the Bell numbers $$b_k (n)b_k (m) \leqslant b_k (n + m) \leqslant \left( {\begin{array}{*{20}c} {n + m} \\ n \\ \end{array} } \right)b_k (n)b_k (m),{\text{ }}\forall n,m \geqslant 0$$ . Then we apply these results to characterization theorems for CKS-space in white noise distribution theory. Kluwer Academic Publishers, 2000 Bell numbers nationallicence log-concavity nationallicence log-convexity nationallicence CKS-space nationallicence characterization theorem nationallicence white noise distribution theory nationallicence Asai Nobuhiro Graduate School of Mathematics, Nagoya University, 464-8602, Nagoya, Japan aut Kubo Izumi Department of Mathematics, Graduate School of Science, Hiroshima University, 739-8526, Higashi-Hiroshima, Japan aut Kuo Hui-Hsiung Department of Mathematics, Louisiana State University, 70803, Baton Rouge, LA, U.S.A. aut Acta Applicandae Mathematica Kluwer Academic Publishers 63/1-3(2000-09-01), 79-87 0167-8019 63:1-3<79 2000 63 10440 https://doi.org/10.1023/A:1010738827855 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1023/A:1010738827855 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Asai Nobuhiro Graduate School of Mathematics, Nagoya University, 464-8602, Nagoya, Japan aut NATIONALLICENCE 700 1- Kubo Izumi Department of Mathematics, Graduate School of Science, Hiroshima University, 739-8526, Higashi-Hiroshima, Japan aut NATIONALLICENCE 700 1- Kuo Hui-Hsiung Department of Mathematics, Louisiana State University, 70803, Baton Rouge, LA, U.S.A aut NATIONALLICENCE 773 0- Acta Applicandae Mathematica Kluwer Academic Publishers 63/1-3(2000-09-01), 79-87 0167-8019 63:1-3<79 2000 63 10440 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer